Methods and apparatus for determining cardiac output and left atrial pressure

ABSTRACT

Method and apparatus are introduced for determining proportional cardiac output (CO), absolute left atrial pressure (LAP), and/or other important hemodynamic variables from a contour of a circulatory pressure waveform or related signal. Certain embodiments of the invention provided herein include the mathematical analysis of a pulmonary artery pressure (PAP) waveform or a right ventricular pressure (RVP) waveform in order to determine beat-to-beat or time-averaged proportional CO, proportional pulmonary vascular resistance (PVR), and/or LAP. The invention permits continuous and automatic monitoring of critical hemodynamic variables with a level of invasiveness suitable for routine clinical application. The invention may be utilized, for example, to continuously monitor critically ill patients with pulmonary artery catheters installed and chronically monitor heart failure patients instrumented with implanted devices for measuring RVP.

BACKGROUND OF THE INVENTION

Cardiac output (CO) is the volume of blood ejected by the heart per unittime, while left atrial pressure (LAP) generally indicates the bloodpressure attained in the left ventricle during the filling phase of thecardiac cycle. CO and LAP are perhaps the two most important quantitiesto be able to monitor in critically ill patients, as they facilitate thediagnosis, monitoring, and treatment of various disease processes suchas left ventricular failure, mitral valve disease, and shock of anycause [36]. For example, a decrease in CO while LAP is rising wouldindicate that the patient is in left ventricular failure, whereas adecrease in CO while LAP is falling would indicate that the patientmight be going into hypovolemic shock.

Several methods are currently available for monitoring CO or LAP. Whileeach of these methods can offer some advantages, as described below, allof the methods are limited in at least one clinically significant way.

The standard methods intended for monitoring CO and LAP in criticallyill patients both involve the use of the balloon-tipped, flow-directedpulmonary artery catheter [47,70]. This catheter also permits continuousmonitoring of pulmonary artery pressure (PAP) and central venouspressure (i.e., right ventricular filling pressure) via fluid-filledsystems attached to external pressure transducers. (However, the mostcommon reason for inserting the pulmonary artery catheter is perhaps inan effort to monitor LAP [46].) CO is specifically estimated accordingto the thermodilution method [17,47]. This method involves injecting abolus of cold saline in the right atrium, measuring temperaturedownstream in the pulmonary artery, and computing the average CO basedon conservation laws. LAP is estimated through pulmonary capillary wedgepressure (PCWP), which is determined by advancing the catheter into abranch of the pulmonary artery, inflating the balloon, and averaging theensuing steady-state pressure [47].

Since PCWP is measured when flow has ceased through the branch, intheory, the PCWP should provide an estimate of LAP. However, the PCWP isnot equal to LAP and is only an approximation [31,44]. In fact, as aresult of a number of technical problems, in practice, the PCWP methodfrequently provides only a poor estimate of LAP. These problems includepartial wedging and balloon over-inflation [38,53], dependence of themeasurement on the wedge catheter position [27,33], and inter-clinicianvariability in interpreting the phasic PCWP measurement [37]. Indeed,the developers of the PCWP method and the balloon-tipped, flow-directedcatheter each reported that they could satisfactorily measure PCWP onlyabout three quarters of the time [58,70]. In 2,711 PCWP measurementattempts made in the ICU, Morris et al reported that only 69% of theseattempts were successful with only 10% of the unsatisfactorymeasurements due to easily correctable damped tracings [53]. Similartechnical problems are also encountered in implementing thethermodilution method in which variations in injectate volume, rate, andtemperature introduce error in the measurement, which is known to be inthe 15-20% range [39,50,68]. Also, the very injection of fluid andballoon inflation poses some risk to the patient [17,34,42]. Perhaps, asa result of these shortcomings, the clinical benefit of the pulmonaryartery catheter has yet to be clearly established (e.g., [64]). Inaddition, a major limitation of the thermodilution and PCWP methods isthat an operator is required. Thus, these important measurements areonly made every few hours.

Alternative methods for monitoring CO include the aortic flow probe,oxygen Fick, dye-dilution, continuous thermodilution, Dopplerultrasound, and thoracic bioimpedance. The aortic flow probe usesultrasound transit-time (or electromagnetic) principles to measureinstantaneous flow [17]. While this method is continuous and veryaccurate, it requires the placement of the flow probe directly aroundthe aorta and is therefore much too invasive for most clinicalapplications. The oxygen Fick method involves simultaneous measurementof central venous and arterial oxygen content of blood and measurementof ventilatory oxygen uptake [17]. Although this method is also highlyregarded in terms of its accuracy, it is too cumbersome for frequentclinical application. The dye dilution method involves the injection ofa dye into the right atrium and serial measurement of the dyeconcentration in blood samples drawn from an arterial catheter [17]. Therelated thermodilution method is generally preferred over this method,because thermodilution requires only one catheter and is less affectedby indicator recirculation. The continuous thermodilution methodinvolves automatic heating of blood in the right atrium via a thermalfilament, measurement of the temperature changes downstream in thepulmonary artery, and computation of average flow via cross correlationand bolus thermodilution principles [17,76]. While this method does notrequire an operator, the temperature changes generated by the thermalfilament must be small to avoid damaging tissue and blood [17]. As aresult, the signal to noise ratio of continuous thermodilution is smallcompared to standard thermodilution, which may render the continuousapproach to be less accurate [78]. Doppler ultrasound methods generallymeasure the Doppler shift in the frequency of an ultrasound beamreflected from the flowing aortic blood [17]. These non-invasive methodsare not commonly employed in critical care medicine, because theyrequire expensive capital equipment and an expert operator to stabilizean external ultrasound transducer [40]. Thoracic bioimpedance involvesmeasuring changes in the electrical impedance of the thorax during thecardiac cycle [17,40]. Although this method is non-invasive andcontinuous, it is too inaccurate for use in critically ill patients dueto excessive lung fluids [11].

Alternative methods for monitoring LAP include direct left heartcatheterization, physical examination, and Doppler imaging. Directcatheterization of the left heart is the gold standard method permittingcontinuous and accurate monitoring of LAP [65]. However, this method istoo invasive and risky for routine clinical application. In a physicalexamination, clinical and radiographic signs of congestion such asrales, third heart sound, prominent jugular vein, and interstitial andalveolar edema are utilized to obtain a qualitative indication ofelevated LAP in patients typically with heart failure [9]. While thisapproach is simple and non-invasive, it is neither continuous nor has itbeen shown to be sensitive to at least changes in PCWP [9,16,69]. InDoppler imaging methods (e.g., color M-mode Doppler, tissue Doppler,pulsed Doppler), parameters such as transmitral and pulmonary venousvelocity profiles are obtained to qualitatively monitor LAP changes orquantitatively monitor PCWP through empirical formulas [21,22,57].Although these techniques may also be non-invasive, they are expensive,can only be used intermittently, and are not specific and may thereforebe inaccurate [2,57, 62].

It would be desirable to be able to accurately monitor both CO and LAPby analysis of a PAP waveform, a right ventricular pressure (RVP)waveform, or other circulatory signals. Thus, unlike the aforementionedmethods, this approach would permit continuous and automatic measurementof these two critically important hemodynamic variables with a level ofinvasiveness suitable for routine clinical use. A continuous andautomatic monitoring approach would be desirable for several reasons.Continuous monitoring of both CO and LAP would be a great advantageduring fluid and pharmaceutical interventions, as the clinician would beable to assess the effects of the interventions and be quickly alertedto possible complications. Continuous monitoring would also provide anearly indication of deleterious hemodynamic events induced by disease(e.g., hypovolemia via simultaneously decreasing CO and LAP). Moreover,automatic monitoring would save precious time in the busy intensive careunit (ICU) and operating room (OR) environments [17] and circumvent theclinically significant problems associated with implementing thestandard measurement methods (see above). Finally, the approach would bea tremendous advantage in the context of remote ICU monitoring (e.g.,[63]) and ambulatory monitoring of PAP or RVP waveforms (in, forexample, heart failure patients) via available implanted devices [1,67].A method for such chronic monitoring of both of these two valuablehemodynamic variables does not otherwise exist.

Many investigators have sought analysis techniques to continuouslymonitor CO from arterial pressure waveforms. Such techniques have beenproposed for over a century [19].

Much of the earlier work assumed that either arterial tree is wellrepresented by a Windkessel model accounting for the compliance of thelarge arteries and the vascular resistance of the small arteries. FIG. 1illustrates the electrical analog of a Windkessel model of the systemicarterial tree. (Because systemic vascular resistance (SVR) is relativelylarge, the model here assumes that the systemic venous pressure (SVP) isnegligible with respect to the systemic arterial pressure (SAP) byvirtue of SVR being referenced to atmospheric pressure rather than SVP.)While techniques based on this simple model generally failed whenapplied to SAP waveforms measured centrally in the aorta (e.g.,[66,72]), Bourgeois et al showed that their technique yielded a quantitythat varied linearly with aortic flow probe CO over a wide physiologicrange [6]. The key concept of their technique is that, according to theWindkessel model, SAP should decay like a pure exponential during eachdiastolic interval with a time constant (τ) equal to the product of SVRand systemic arterial compliance (AC). Since AC is nearly constant overa wide pressure range and on the time scale of months [5,26,60], COcould then be measured to within a constant scale factor by dividing thetime-averaged SAP with τ. Thus, the technique of Bourgeois et alinvolves fitting an exponential function to each diastolic interval of aSAP waveform to measure τ (FIG. 1).

Bourgeois et al were able to validate their technique with respect tocentral SAP waveforms, because the diastolic interval of these waveformscan sometimes resemble an exponential decay following incisura (FIG. 2a). These investigators identified a precise location in the thoracicaorta as the optimal site in the canine for observing an exponentialdiastolic decay. However, central SAP is rarely measured clinically dueto the risk of blood clot formation and embolization. Moreover,exponential diastolic decays are usually not apparent in eitherperipheral SAP waveforms (FIG. 2 b), which may be measured via minimallyinvasive radial artery catheterization, or PAP waveforms (FIG. 2 c).Indeed, Bourgeois et al acknowledged that exponential diastolic decaysare obscured in peripheral SAP waveforms [5]. Moreover, after Engelberget al suggested that the pulmonary arterial tree be represented by aWindkessel model in which the small pulmonary vascular resistance (PVR)is referenced to LAP (electrical analog in FIG. 3) [18], Milnor et alattempted to fit an exponential function to each diastolic decayinterval of PAP waveforms minus average LAP in man and reported that allof the waveforms were not adequate for doing so [51]. Subsequently,Tajimi et al reported that they were not able to identify a location inthe canine or human pulmonary artery in which exponential diastolicpressure decays were consistently visible [71]. The reason is that thesystemic and pulmonary arterial trees are not simply lumped systems likethe Windkessel model suggests but rather complicated distributed systemswith impedance mismatches throughout due to vessel tapering,bifurcations, and caliber changes. The diastolic (and systolic)intervals of peripheral SAP and PAP waveforms are therefore corrupted bycomplex wave reflections occurring at each and every site of impedancemismatch. Moreover, inertial effects also contribute to obscuringexponential diastolic decays, especially in the low-pressure pulmonaryarterial tree [55]. Thus, the technique of Bourgeois et al cannot beapplied to clinically measurable peripheral SAP and PAP waveforms.

More recently, investigators have attempted to monitor CO fromperipheral SAP waveforms. Techniques based on an adaptive aorta model,which require SAP waveforms measured at both the carotid and femoralarteries have been proposed [59,73]. However, catheters are usually notplaced for prolonged periods of time at either of these sites in ICUs,ORs, or recovery rooms due to safety considerations. A technique hasbeen introduced that is based on an empirically derived formulainvolving the calculation of the derivative of the ABP waveform [23].However, in order to mitigate the corruptive effects of wave reflectionson the derivative calculation, this technique also requires twoperipheral ABP measurements, one of which is obtained from the femoralartery. Learning techniques requiring training data sets consisting ofsimultaneous measurements of CO and SAP waveforms have also beensuggested [8,24,48]. However, these techniques were only demonstrated tobe successful in central ABP waveforms or over a narrow physiologicrange. Moreover, learning techniques could only be successful providedthat the available training set of patient data reflected the entirepatient population. Finally, Wesseling et al [3,74] and Linton et al[41] have proposed techniques requiring only the analysis of a singleradial SAP waveform. However, Linton et al only showed that theirtechnique was accurate over a narrow physiologic range, and severalstudies have demonstrated limitations of the technique of Wesseling etal (e.g., [20,29]).

A technique for continuous CO monitoring from peripheral SAP waveformshas been the recent focus of interest, because it is minimally invasiveor possibly even non-invasive (e.g., [24]). However, even if such atechnique were introduced with sufficient accuracy, the more invasivepulmonary artery catheters would still be used to be able to measureleft and right ventricular filling pressures. Four investigators havetherefore previously attempted to monitor CO continuously by analysis ofPAP waveforms [10,15,71,77]. In this way, ventricular filling pressuresand continuous CO could be measured with a single catheter. Theseinvestigators essentially employed analysis techniques that werepreviously applied to SAP waveforms. Their results showed that thetechniques could estimate CO during cardiac interventions but notvascular interventions (e.g., volume infusion). Moreover, even thoughLAP is also a significant determinant of PAP and should therefore bereflected in the PAP waveform, none of these techniques included a meansto monitor LAP. In fact, similar to the suggestion of Engelberg et al,the technique of Cibulski et al actually required an additional LAPmeasurement for monitoring CO [10].

The common feature of all of the aforementioned techniques formonitoring CO from continuous SAP or PAP is that the waveform analysisis employed only over time scales within a cardiac cycle. Because of thecorruptive effects of highly complex waves at these time scales, thetechniques were limited in that they 1) could only be applied to highlyinvasive central SAP waveforms in which the complex wave reflections maybe attenuated; 2) necessitated multiple peripheral SAP waveformmeasurements (which are rarely obtained clinically); 3) required anexhaustive training data set, and/or 4) are accurate only over a narrowphysiologic range or only during cardiac interventions. However, theconfounding effects of wave as well as inertial phenomenon are known todiminish with increasing time scale [55]. Based on thisunder-appreciated concept, Mukkamala et al introduced a technique tomonitor CO by analyzing a single arterial pressure waveform (measured atany site in the systemic or pulmonary arterial trees) over time scalesgreater than a cardiac cycle [54]. They evaluated their technique withrespect to peripheral SAP waveforms in swine and their results showedexcellent agreement with aortic flow probe measurements over a widephysiologic range [54]. While this technique may permit continuous andaccurate monitoring of CO with a level of invasiveness suitable forroutine clinical application, it does not provide a convenient means tomonitor LAP.

Some investigators have attempted to monitor LAP or PCWP by analysis ofblood pressure waveforms. Shortly after the introduction of thepulmonary artery catheter, researchers studied the end-diastolic PAP asa continuous index of LAP [28,31]. However, this simple technique is notas accurate as the PCWP method [31] (see below) and becomes unreliablewhen the rate of drainage of blood from the pulmonary artery into thepulmonary capillaries is slow. Thus, for example, it is well known thatthe end-diastolic PAP is not an acceptable index of LAP in patients withpulmonary vascular disease [28]. More recently, a learning technique hasbeen proposed to monitor PCWP from a PAP waveform using an artificialneural network trained on a database of PCWP measurements and PAPwaveforms [14, 78]. However, this technique was shown to be ineffectivewhen the network was trained on one set of subjects and tested on adifferent set of subjects [14]. Finally, techniques have been proposedin which trained regression equations predict LAP or PCWP fromvariations in parameters of the SAP or plethysmographic waveform (e.g.,systolic pressure, pulse pressure) in response to the Valsalva maneuveror mechanical positive pressure ventilation [45,49,65,75]. While thesetechniques may be minimally invasive or non-invasive, they are eithernot continuous or applicable to subjects breathing spontaneously.Moreover, since SAP and related signals are also due to ventricular andarterial functionality, these techniques do not provide a specificmeasure of LAP or PCWP and may therefore be inaccurate [13].

Thus, a technique is needed that accurately monitors both CO and LAP byanalysis of a PAP waveform, a RVP waveform, or other circulatorysignals. Such a technique could be utilized, for example, tocontinuously monitor critically ill patients instrumented with pulmonaryartery catheters and chronically monitor heart failure patientsinstrumented with implanted devices.

SUMMARY OF THE INVENTION

The present invention involves the mathematical analysis of the contourof a PAP waveform, a RVP waveform, or another circulatory signal inorder to determine average LAP, average SVP, proportional CO,proportional PVR, proportional SVR, and/or other clinically importanthemodynamic variables. In various embodiments, the methods of theinvention may be employed to make individual measurements of averageLAP, average SVP, proportional CO, proportional PVR, proportional SVR,and/or other clinically important hemodynamic variables at one or moretime instances or may be employed for continuous monitoring of one ormore of these variables.

The mathematical analysis of a circulatory contour is defined here tocomprise an examination of the temporal variations in the signal. Thetemporal variations in the signal can be examined within a singlecardiac cycle, between different cardiac cycles, or both. Simplymonitoring the signal at the same time instance relative to aphysiologically significant event, such as the end of diastole or theend of systole, in multiple cycles does not constitute performing amathematical analysis of a circulatory contour. Furthermore, the abovedefinition is intended to exclude trivial examinations of the temporalvariations in the signal that solely provide an estimate of the timeinstance of such a physiologically significant event (e.g., zerotemporal derivative of the signal). Thus, for example, monitoring LAPvia the end-diastolic PAP does not constitute a mathematical analysis ofa circulatory contour even if the time instance of the end of diastoleis determined via the zero temporal derivative of the PAP waveform(i.e., the time point at which the derivative of the PAP waveform iszero), since in effect this examination indicates PAP at particularinstances of time.

Typically, an examination of the temporal variations in the signalinvolves the analysis of at least two time instances of the signalwithin a single cardiac cycle or at least two time instances of thesignal within at least two cardiac cycles. In certain embodiments of theinvention, an examination of the temporal variations in the signalinvolves the analysis of an interval of the signal within a cardiaccycle (e.g., fraction of the diastolic interval greater than 25%) orencompassing multiple cardiac cycles (e.g., at least 30 cardiac cycles).In certain embodiments of the invention, an examination of the temporalvariations in the signal involves the analysis of an interval of thesignal within individual cardiac cycles (e.g., systolic ejectioninterval) over a period encompassing multiple cardiac cycles (e.g., atleast 30 cardiac cycles). In various embodiments of the invention theaverage values for hemodynamic variables computed as described hereinmay be averages over a single beat or over multiple beats.

In certain embodiments of the invention, the mathematical analysis of acirculatory contour involves the use of the Windkessel model of FIG. 3to represent the slow dynamical properties of the pulmonary arterialtree.

In one such embodiment, proportional CO, average LAP, and otherhemodynamic variables such as proportional PVR are determined bymathematical analysis of one or more individual diastolic decayinterval(s) of a PAP waveform. For example, in certain embodiments ofthe invention, a set of basis functions, one or more of which mayinclude a constant term or may be a constant term, are fitted to eachdiastolic decay interval of PAP to respectively determine the dynamicalproperties of the pulmonary arterial tree and the average LAP. Any basisfunctions known in the art may be employed. For example, realexponentials, complex exponentials, and/or polynomials (e.g., over aspecific time interval) can be used.

In certain embodiments of the invention, complex exponential functionsand a constant term are fitted to each diastolic decay interval of PAPto respectively determine the dynamical properties of the pulmonaryarterial tree and the average LAP (FIG. 4). Any number of complexexponential functions (usually an odd number) may be utilized in thefitting procedure to account for wave, inertial, and Windkessel effectsin the pulmonary arterial tree. The fitting procedure may be performedusing any method known in the art and applied to, for example, theentire PAP downstroke (i.e., from maximum to minimum pressure; FIG. 4).Then, the Windkessel time constant τ is determined by, for example,extrapolating the computed exponential functions to low-pressure valuesand fitting a mono-exponential function to the extrapolated pressurevalues for which the faster wave and inertial effects have dissipated orvanished (FIG. 4). The average LAP is determined by, for example,computing the mean value of the resulting constant term (i.e., theconstant term that the mono-exponential function approachesasymptotically) over any number of beats, while proportional CO isdetermined, for example, by computing the mean value of the resultingPAP-LAP (i.e., the mean value resulting from subtracting the LAP fromPAP) over any number of beats and dividing this quantity by the meanvalue of the corresponding τ.

In other embodiments of the invention, the set of basis functions is aset of decaying polynomial functions, at least one of which is orincludes a constant term. The polynomials may be extrapolated to lowpressure values, and a single exponential can then be fitted to the lowpressure values to determine the time constant. Alternatively, the basisfunctions can be extrapolated to low pressure values, and the timeconstant may be measured from the extrapolated pressure values by anyother method known in the art (e.g., the time duration required for theextrapolated pressure to drop by a predetermined amount, such as by afactor of e⁻¹ (36.79%). It will be appreciated that other methods fordetermining average LAP from the unknown constant, e.g., withoutperforming extrapolation, may also be used.

In another such embodiment, proportional CO, average LAP, and/or otherhemodynamic variables such as proportional PVR are determined bymathematical analysis of all temporal variations in a PAP waveformincluding those occurring over time scales greater than a cardiac cyclein which the confounding effects of wave and inertial phenomena areattenuated [55]. For example, a segment of a PAP waveform of durationgreater than a cardiac cycle (e.g., ranging from 30 seconds toten-minutes) is fitted according to the sum of an unknown constant termrepresenting average LAP and the convolution of an unknown impulseresponse characterizing the dynamical properties of the pulmonaryarterial tree with a known signal reflecting each cardiac contraction.The cardiac contractions signal may be established from the PAP waveformand possibly other physiologic signals by, for example, forming animpulse train in which each impulse is located at the onset of eachcardiac contraction and has area equal to an arbitrary constant, theensuing pulse pressure (x(t) in FIG. 5), or any other value based on thePAP pulse (e.g., the ensuing pulse pressure determined after lowpassfiltering the PAP waveform to attenuate the wave and inertial effects orthe value of PAP at any defined time instance in the cardiac cyclerelative to the onset of a cardiac contraction with or without respectto the value of PAP at the cardiac contraction onset). The unknownconstant term and impulse response are determined so as to permit thebest fit of the PAP waveform segment according to any of the methodsknown in the art. The process of determining the unknown constant termand impulse response that provide the best fit, or an acceptable fit, toa waveform or portion thereof is referred to as “fitting” herein. Theresulting pulmonary arterial tree impulse response is defined torepresent the PAP-LAP response to a single, solitary cardiac contraction(h(t) in FIG. 5). The Windkessel time constant X of the pulmonaryarterial tree is then determined from this impulse response by fittingan exponential to its tail end once the faster wave and inertial effectshave vanished (h(t) in FIG. 5). Finally, proportional CO is determinedas the difference between the mean value of the analyzed PAP waveformsegment and average LAP (i.e., the resulting constant term) divided bythe resulting τ. Alternatively, when the area of each impulse of thecardiac contractions signal is set to an arbitrary constant,proportional CO may be determined as the product of the peak value ofthe pulmonary arterial tree impulse response and the average heart rate.

The above aspects of the present invention may be utilized, for example,in patients instrumented with pulmonary artery catheters in ICUs, ORs,and recovery rooms. In such applications, the continuous, proportionalCO, PVR, and/or X values may be conveniently calibrated to continuous,absolute CO and PVR values with a single thermodilution measurement. Itwill be appreciated that certain embodiments of the invention provideabsolute average LAP without requiring calibration based on athermodilution measurement.

Preferred methods of the invention for determining proportional CO,average LAP, proportional PVR, and/or other clinically importanthemodynamic variables by performing a mathematical analysis of a PAPwaveform, e.g., as described above, do not entail the use of a neuralnetwork. Certain preferred methods of the invention also do not make useof a set of training data. For example, the methods of the invention fordetermining proportional CO, average LAP, proportional PVR, and/or otherclinically important hemodynamic variables do not require makingconventional measurements of a subject's PCWP (e.g., by advancing apulmonary artery catheter into a branch of the pulmonary artery,inflating the balloon, and averaging the ensuing steady-state pressure)as a prerequisite to the monitoring of proportional CO, average LAP,and/or other clinically important hemodynamic variables in the subject.The methods of the invention for determining proportional CO, averageLAP, proportional PVR, and/or other clinically important hemodynamicvariables also do not require a training data set comprisingconventional measurements of PCWP in one or more subjects in order tomonitor, according to the methods of the invention, proportional CO,average LAP, and/or other clinically important hemodynamic variable(s)in the same subjects or different subjects.

It will thus be appreciated that the methods described above provide ameans of determining LAP and/or other clinically important hemodynamicvariables based on first principles rather than estimating ordetermining PCWP and then using the estimated or determined PCWP as anapproximation to LAP. In preferred embodiments, the methods andapparatus of the invention are therefore not adapted specifically fordetermining PCWP from a PAP waveform based on a correlation betweenmeasured values of the PAP and PCWP but instead determine LAP withoutrelying on a correlation between measured PAP and PCWP. As mentionedabove, in practice PCWP is usually measured primarily in order toprovide an estimate of LAP even though such an estimate suffers from anumber of limitations noted above. Thus, the methods of the inventionoffer an advantage over prior art methods that employ either a measuredor estimated PCWP value as an estimate for LAP. However, if desired, theLAP determined according to the methods of the invention could be usedto approximately determine PCWP. That is, LAP determined according tothe methods of the invention could either be used directly as anapproximate of PCWP, or the determined LAP could be adjusted in order toprovide an improved estimate of PCWP. For example, in one embodiment,such an adjustment is accomplished through a training data setcomprising PAP waveforms and conventional PCWP measurements. Morespecifically, LAP is determined as described above with respect to thePAP waveforms in the training data set, and a model (e.g., linear ornonlinear regression) is developed according to any of the methods knownin the art to predict the corresponding PCWP measurements in thetraining data set. Then, PCWP is subsequently estimated from a PAPwaveform by first determining LAP according to the methods of theinvention and then applying the trained model to the determined LAP topredict PCWP.

In yet another such embodiment, proportional CO, average LAP, and/orother hemodynamic variables such as proportional PVR are determined bymathematical analysis of the contour of a RVP waveform. The RVP waveformis preferably obtained from patients without stenosis of the pulmonicvalve so that the RVP waveform may be regarded as equivalent to theunobserved PAP waveform during each systolic ejection interval. Thesystolic ejection intervals of the RVP waveform are identified by anymethod known in the art, e.g., using a phonocardiogram or otherphysiologic signals. For example, the beginning of systole for each beatmay be determined as the time of the maximum temporal derivative of eachRVP pulse [61], while the end of systole for each beat may be identifiedas the time of the peak value of each RVP pulse (y(t) in FIG. 6 a).Then, with this “incomplete” PAP waveform, also referred to as a“partial” PAP waveform herein, the clinically significant hemodynamicvariables may be determined similarly to the embodiments describedabove. For example, the systolic ejections intervals of a segment of aRVP waveform of duration greater than a cardiac cycle (e.g., rangingfrom 30 seconds to ten-minutes) are fitted according to the sum of anunknown constant term representing the average LAP and the convolutionof an unknown impulse response representing the dynamical properties ofthe pulmonary arterial tree with a known and complete cardiaccontractions signal as defined above. The impulse response and constantterm that provide the best fit of the systolic ejection intervals of theRVP waveform segment (based on any method known in the art) are thenutilized to determine average LAP and the Windkessel time constant τ ofthe pulmonary arterial tree as described above (FIG. 6 a). Next, thecomplete PAP waveform segment (including diastolic intervals) isconstructed by the convolution between the resulting impulse responseand the complete cardiac contractions signal (z(t) in FIG. 6b) plus theresulting average LAP. Finally, proportional CO is determined either asthe difference between the mean value of the constructed PAP waveformsegment and the resulting average LAP divided by the resulting τ or theproduct of the peak value of the pulmonary arterial tree impulseresponse and the average heart rate. This particular aspect of theinvention may be utilized, for example, in patients instrumented withimplanted devices capable of monitoring a RVP waveform [1].

The method (and apparatus) for constructing a PAP waveform containingboth systolic and diastolic intervals based on the systolic ejectionintervals of the RVP waveform is an aspect of this invention. The PAPwaveform constructed according to the method of the invention can befurther analyzed for any of a variety of purposes including, but notlimited to, determining average LAP, proportional CO, or proportionalPVR as described herein.

In another embodiment of the invention, the mathematical analysis of acirculatory contour involves the use of a Windkessel model analogous toFIG. 3 to represent the slow dynamical properties of the systemicarterial tree. In one such embodiment, the above mathematical proceduresdescribed in the context of PAP waveform analysis are applied to a SAPwaveform (or a related waveform such as a photoplethysmography signal)measured at any site in the systemic arterial tree. Similarly, the abovemathematical procedures described in the context of RVP waveformanalysis may be applied to a left ventricular pressure (LVP) waveform inwhich aortic stenosis is absent. In such embodiments, the dynamicalproperties of the systemic arterial tree and the average SVP (throughthe constant term) are determined. Alternatively, SVP may be regarded asnegligible with respect to SAP (i.e., left panel of FIG. 1) and only thedynamical properties of the systemic arterial tree are determined. Thedetermined dynamical properties may then be analogously utilized todetermine proportional CO and proportional SVR. These particular aspectsof the invention may be utilized, for example, in ICUs, ORs, andrecovery rooms in which radial artery catheters are routinely employedor at home, in emergency rooms, and in the ward in which non-invasiveSAP waveforms (e.g., finger-cuff photoplethysmography [32], arterialtonometry [35]) could easily be obtained. The continuous, proportionalCO and τ values may be conveniently calibrated to continuous, absoluteCO and SVR values with a single, absolute measurement of CO.

The method for constructing a SAP waveform containing both systolic anddiastolic intervals based on the systolic ejection intervals of the LVPwaveform is an aspect of this invention. The SAP waveform constructedaccording to the method of the invention can be further analyzed for anyof a variety of purposes including, but not limited to, determining SVP,proportional CO, or proportional SVR as described herein.

In any of these embodiments, the constant term represents the asymptoticvalue of the waveform were cardiac contractions to cease. Thus theinvention provides a method for determining the value of a hemodynamicvariable based on a PAP, RVP, SAP, or LVP waveform, comprising steps of(i) performing a mathematical analysis to determine the asymptotic valueof the waveform were cardiac contractions to cease and (ii) selectingthe asymptotic value as the value for the hemodynamic variable. Incertain embodiments of the invention the waveform is a PAP waveform andthe hemodynamic variable is average LAP. In certain embodiments of theinvention the waveform is a RVP waveform and the hemodynamic variable isaverageLAP. In certain embodiments of the invention the waveform is aSAP waveform and the hemodynamic variable is average SVP. In certainembodiments of the invention the waveform is a LVP waveform and thehemodynamic variable is average SVP. Any of the methods may furthercomprise determining the value of proportional CO, proportional PVR, orproportional SVR.

The invention further comprises apparatus for use in performing any oneor more of the methods of the invention.

The invention further provides computer-executable process steps storedon a computer-readable medium for performing one or more methods of theinvention. The invention further provides methods of monitoring asubject comprising performing one or more of the inventive methods on aPAP waveform, RVP waveform, SAP waveform or related signal, or LVPwaveform obtained from the subject. The methods may further comprise thestep of obtaining such a waveform from the subject. The methods mayfurther comprise the step of administering a therapy to the subject, ormodifying the subject's therapy, based on one or more hyemodynamicvariables obtained according to the methods.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1: Previous analysis technique for monitoring cardiac output (CO)from a systemic arterial pressure (SAP) waveform [6]. According to theWindkessel model of the systemic arterial tree (electrical analog to theleft), SAP should decay like a pure exponential during each diastolicinterval with a time constant (τ) equal to the product of the systemicvascular resistance (SVR) and the nearly constant arterial compliance(AC). Thus, this technique involves first fitting an exponentialfunction to each diastolic interval of the SAP waveform to determine τ(right) and then dividing the time-averaged SAP (MAP) with τ to estimateproportional CO.

FIG. 2: Swine arterial pressure waveforms measured a) centrally in theaorta, (b) peripherally in the radial artery, and (c) in the mainpulmonary artery. The diastolic intervals of the central systemicarterial pressure (SAP) waveform resemble exponential decays; however,the measurement of central SAP is too invasive for routine clinicalapplication. In contrast, exponential diastolic decays are not visiblein the peripheral SAP and pulmonary artery pressure (PAP) waveforms,which are heavily corrupted by fast wave reflections and inertialeffects. Thus, the previous technique in FIG. 1 cannot be applied toclinically measurable peripheral SAP and PAP waveforms.

FIG. 3: Electrical analog of the Windkessel model of the pulmonaryarterial tree. One embodiment of the mathematical analysis utilizes thismodel to represent the slow dynamical behavior of the pulmonary arterialtree. For example, if pulsatile activity suddenly ceased, this modelpredicts that pulmonary artery pressure (PAP) would eventually decaylike a pure exponential with a Windkessel time constant T equal to theproduct of the pulmonary vascular resistance (PVR) and the nearlyconstant pulmonary arterial compliance (PAC) as soon as the faster waveand inertial effects vanished (see FIG. 2). The model also indicatesthat the exponential pressure decay would equilibrate towards the leftatrial pressure (LAP), which significantly contributes to PAP due to therelatively small PVR.

FIG. 4: One embodiment in which the mathematical analysis is applied tothe individual diastolic decay intervals of a pulmonary artery pressure(PAP) waveform (y(t)). Multiple, complex exponential functions and aconstant term are fitted to each diastolic decay interval of y(t) torespectively determine the dynamical properties of the pulmonaryarterial tree and the average left atrial pressure (LAP). Then, theWindkessel time constant τ of the pulmonary arterial tree (see FIG. 3)is determined by extrapolating the computed exponential functions tolow-pressure values and fitting a mono-exponential function to theextrapolated pressure values for which the faster wave and inertialeffects have vanished. Finally, proportional cardiac output (CO) isdetermined via Ohm's law (overbar indicates time averaging). Thisintra-beat embodiment of the mathematical analysis may be capable ofmonitoring beat-to-beat changes in CO and LAP.

FIG. 5: One embodiment in which the mathematical analysis is applied toall temporal variations in a pulmonary artery pressure (PAP) waveform(y(t)) including those occurring over time scales greater than a cardiaccycle. Average left atrial pressure (LAP) and the response of PAP minusaverage LAP to a single, solitary cardiac contraction (h(t)) aresimultaneously estimated by fitting a segment of y(t). Then, theWindkessel time constant r of the pulmonary arterial tree (see FIG. 3)is determined by fitting a mono-exponential function to the tail end ofh(t) once the faster wave and inertial effects have vanished. Finally,proportional cardiac output (CO) is computed via Ohm's law (overbarindicates time averaging). This inter-beat embodiment of themathematical analysis is expected to be able to accurately determineaverage r and LAP, because beat-to-beat PAP variations are hardlyconfounded by wave and inertial phenomena [55]. PP is pulse pressure; R,the time of the onset of upstroke of each PAP pulse; j, the j^(th) beat;x(t), an impulse train signal representing cardiac contractions.

FIG. 6: One embodiment in which the mathematical analysis is applied toa right ventricular pressure (RVP) waveform (y(t)) obtained frompatients without pulmonic valve stenosis. (a) The systolic ejectionintervals of a RVP waveform segment are identified and regarded asequivalent to the unobserved pulmonary artery pressure (PAP). Then,average left atrial pressure (LAP) and the response of PAP minus averageLAP to a single, solitary cardiac contraction (h(t)) are simultaneouslyestimated by fitting the incomplete segment of y(t). Next, theWindkessel time constant τ of the pulmonary arterial tree (see FIG. 3)is determined by fitting a mono-exponential function to the tail end ofh(t) once the faster wave and inertial effects have vanished. (b)Finally, the complete PAP-LAP waveform segment (including diastolicintervals) is constructed (z(t)), and proportional cardiac output (CO)is determined via Ohm's law (overbar indicates time averaging). PP ispulse pressure; R, the time of the onset of upstroke of each PAP pulse;j, the j^(th) beat; x(t), an impulse train signal representing cardiaccontractions.

DETAILED DESCRIPTION OF CERTAIN EMBODIMENTS OF THE INVENTION

The present invention is based at least in part on the recognition thatproportional CO, average LAP, and/or other hemodynamic variables arereflected in the contour of a PAP waveform, a RVP waveform, or othercirculatory signals. The invention therefore involves the mathematicalanalysis of the contour of a circulatory signal so as to decipher, andthereby determine, e.g., continuously, these clinically significanthemodynamic variables. The mathematical analysis of a circulatorycontour is specifically defined here to comprise an examination of thetemporal variations in the signal. Thus, for example, monitoring LAP viathe end-diastolic PAP does not constitute a mathematical analysis of acirculatory contour, since this examination simply indicates PAP atparticular instances of time (end-diastole).

In one embodiment of the invention, the mathematical analysis of acirculatory contour is based on the Windkessel model of the pulmonaryarterial tree shown in FIG. 3. (Although PVR is known to be nonlinearover a wide pressure range [25], it may be approximated as linear overthe more narrow range of PAP variations that are considered by variousembodiments of the present invention.) According to this model, PAPshould decay like a pure exponential during each diastolic interval witha time constant (τ) equal to the product of PVR and the pulmonary AC(PAC). The model further indicates that the exponential pressure decayshould equilibrate towards average LAP rather than zero pressure. Thus,the Windkessel model of FIG. 3 suggests that both τ and average LAP maybe determined from a PAP waveform by fitting a single exponentialfunction plus a constant term to each of its diastolic intervals.Moreover, since PAC may be nearly constant over a wide pressure range,CO may also be determined to within a constant scale factor by dividingthe time-averaged PAP-LAP with τ. However, the invention encompasses therecognition that this simple fitting procedure is not generally valid inpractice, because exponential diastolic decays are usually obscured inphysiologic PAP waveforms by complex wave and inertial effects (FIG. 2c). To determine the Windkessel parameters (τ and average LAP) from aPAP waveform, the invention further encompasses the recognition that theconfounding wave and inertial dynamics are faster than the exponentialWindkessel dynamics [55]. This implies that if pulsatile activity wereto abruptly cease, then PAP would eventually decay like a pureexponential and ultimately equilibrate to the LAP once the faster waveand inertial dynamics vanished. Thus, in this embodiment, the Windkesselmodel of FIG. 3 is specifically used to represent only the slowdynamical properties of the pulmonary arterial tree.

In one such embodiment, proportional CO, average LAP, and/or otherhemodynamic variables are determined by mathematical analysis of eachindividual diastolic decay interval of a PAP waveform. For example,basis functions, e.g., multiple, complex exponential functions and aconstant term are fitted to each diastolic decay interval of PAP torespectively determine the dynamical properties of the pulmonaryarterial tree and the average LAP (FIG. 4). Then, the Windkessel timeconstant τ of the pulmonary arterial tree may be determined byextrapolating the computed exponential functions to low-pressure valuesand fitting a mono-exponential function to the extrapolated pressurevalues for which the faster wave and inertial effects have dissipated(FIG. 4). Finally, proportional CO may be determined over any number ofbeats via Ohm's law.

FIG. 4 illustrates the details of this embodiment of the mathematicalanalysis, which is applied to a digitized PAP waveform sampled at, forexample, 90 Hz. It will be appreciated that different sampling ratescould be used. This embodiment is specifically employed in four steps.

First, the diastolic decay interval of each PAP pulse is identified asthe entire downstroke (i.e., from maximum to minimum pressure) or by anyother method known in the art. For example, each diastolic decayinterval may be identified with a simultaneous phonocardiogrammeasurement. As another example, each diastolic decay interval may bedetermined according to any formula based on the cardiac cycle length(T) such as the well-known Bazett formula (T−0.3√{square root over (T)})[4]. Each cardiac cycle length may be determined from the PAP waveformand/or other simultaneously measured physiologic signals such as asurface ECG measurement.

Second, any number of complex exponential functions and a constant termare fitted to each of the identified diastolic decay intervals of thePAP waveform (v(t)). The estimated constant term represents the averageLAP, while the estimated complex exponential functions characterize thewave, inertial, and Windkessel dynamical properties of the pulmonaryarterial tree. The “best” fit complex exponential functions and constantterm may be specifically estimated based on the following output errorequation with constant term c and unit impulse (δ(t)) input:

$\begin{matrix}{{{h(t)} = {{\sum\limits_{k = l}^{n}{a_{k}{h\left( {t - k} \right)}}} + {\sum\limits_{k = l}^{n}{b_{k}{\delta \left( {t - k} \right)}}}}}{{y(t)} = {c + {h(t)} + {{e(t)}.}}}} & (1)\end{matrix}$

Here, e(t) is the unmeasured residual error, the pair of parameters{a_(k), b_(k)} completely specify each complex exponential function, nindicates the number of complex exponential functions, and h(t)represents the temporal evolution of the complex exponential functionscollectively for all time. For a given value of n, the parametersincluding c are estimated from the diastolic decay intervals of y(t)through the least-squares minimization of the residual error [43]. Thisoptimization problem may be solved through a numerical search (e.g.,Gauss Newton method) or the Stieglitz-McBride iteration [43].Alternatively, the parameters may be more conveniently, but notoptimally, estimated using other methods known in the art such asProny's method and Shank's method [7,56]. A pre-determined number ofcomplex exponential functions (usually an odd number such as n=1, 3 or5) may be utilized in the estimation procedure. Alternatively, anoptimal number of complex exponential functions may be determinedaccording to any of the methods known in the art that penalize forunnecessary parameters (e.g., root-normalized-mean-squared-error (RNMSE)threshold) [43]. Prior to this estimation procedure, y(t) may be lowpassfiltered in order to attenuate the complex wave and inertial effects.Note that any other parametric model with complex exponential basisfunctions (e.g., autoregressive exogenous input (ARX) model [43]) or anyother functions (e.g., polynomials) may be employed in variousembodiments of the invention to represent the diastolic decay intervalof each PAP pulse, and any other minimization criterion (e.g., absoluteerror) may be utilized to determine the “best” fitting functions andconstant term. It will be appreciated that it is not necessary to selectthe parameters that in fact result in minimizing the error, althoughthese parameters may provide the most accurate results. Instead, theparameters can be selected such that the error is below a predeterminedvalue (e.g., 110-120% of the minimum error), in which the predeterminedvalue is specifically selected so as to achieve an acceptable accuracyfor the purposes at hand.

Third, the estimated complex exponential functions are extrapolated tolow-pressure values. This step is achieved by recursively solving forh(t) (until it is effectively zero) based on the estimated parameters{â_(k),{circumflex over (b)}_(k)} as follows:

$\begin{matrix}{{h(t)} = {{\sum\limits_{k = l}^{n}{{\hat{a}}_{k}{h\left( {t - k} \right)}}} + {\sum\limits_{k = l}^{n}{{\hat{b}}_{k}{{\delta \left( {t - k} \right)}.}}}}} & (2)\end{matrix}$

The Windkessel time constant τ of the pulmonary arterial tree is thendetermined over the interval of h(t) preferably over a time interval inwhich the faster wave and inertial effects have become minimal. Forexample, as shown in FIG. 4, the contribution of the faster wavereflections and inertial effects becomes minimal within one second ofthe peak value of h(t). Following this time interval, h(t) may beaccurately approximated as a mono-exponential. Thus, in certainpreferred embodiments of the invention, the selected time intervalbegins approximately 0.75 seconds following the time of maximum h(t) or,more preferably, approximately one second following the time of maximumh(t). For example, it has been found that a time interval of 1 to 2seconds following the time of maximum value of h(t) is suitable. Longertime intervals may also be used. Typically the appropriate time intervalis predetermined, but, in certain embodiments of the invention, it maybe selected as the measurements are being made. The determination of τis achieved based on the selected interval of h(t) according to thefollowing single exponential equation:

h(t)=Ae ^(−t/τ) +w(t).   (3)

The parameters A and τ may be estimated according to any procedure knownin the art. For example, the parameters may be determined by theclosed-form, linear least squares solution after log transformationofh(t). Alternatively, τ may be identified as the largest real valuedtime constant of the estimated complex exponential functions. Note thatthis step is unnecessary when only a single exponential function isutilized to fit the diastolic decay interval of the PAP pulse.

Finally, the average LAP is determined as the mean or median value ofthe estimated constant term ĉ over any number of beats, while CO isdetermined to within a constant scale factor equal to 1/PAC as thedifference between the mean value of PAP over any number of beats andthe corresponding average LAP divided by the mean or median value of thecorresponding τ. Note that PVR is also trivially determined to within aconstant scale factor via τ. The proportional, continuous CO and PVRestimates may be calibrated, if desired, with a single, absolutemeasurement of CO. Such a calibration may be conveniently implementedwith the bolus thermodilution method, if a pulmonary artery catheterwere employed. Otherwise, if an independent CO measurement isunavailable, then a nomogram may be utilized, if desired, to calibratethe proportional estimates.

Alternatively, proportional CO, average LAP, and other hemodynamicvariables are determined by mathematical analysis of all temporalvariations in a PAP waveform including those occurring over time scalesgreater than a cardiac cycle in which the confounding effects of waveand inertial phenomena are attenuated [55]. For example, average LAP andthe response of PAP minus average LAP to a single, solitary cardiaccontraction (h(t) in FIG. 5) are simultaneously estimated by fitting aPAP waveform segment of duration greater than a cardiac cycle (e.g., 30seconds to 10 minutes). Then, the Windkessel time constant s of thepulmonary arterial tree is determined by fitting a mono-exponentialfunction to the tail end of h(t) once the faster wave and inertialeffects have vanished (FIG. 5). Finally, proportional CO may be computedby dividing the time-averaged PAP-LAP with T. Without wishing to bebound by any theory, while an intra-beat embodiment will likely providebetter temporal resolution (e.g., ability to detect beat-to-beathemodynamic changes), an inter-beat embodiment is expected to be moreaccurate in terms of estimating average hemodynamic values as thebeat-to-beat PAP variations are much more reflective of Windkesseldynamics than confounding wave and inertial phenomena (i.e., a largersignal-to-“noise” ratio) [55].

FIG. 5 illustrates the details of this embodiment of the mathematicalanalysis, which is applied to a segment of a digitized PAP waveform(sampled at, e.g., 90 Hz) of duration greater than a cardiac cycle(e.g., 30 seconds to ten minutes). It will be appreciated that differentsampling rates could be used. This embodiment is specifically employedin four steps.

First, a cardiac contractions signal is constructed by forming animpulse train in which each impulse is located at the onset of upstrokeof a PAP pulse (R) and has an area equal to the ensuing pulse pressure(PP). PP is determined, for example, as the maximum PAP value minus thePAP value at the onset of upstroke. Alternatively, each impulse may beplaced at the R-wave of a simultaneous surface ECG measurement, and/orthe area of each impulse may be set to an arbitrary constant or anyother value based on the PAP pulse. As another alternative, pulses ofany shape (e.g., rectangular) with duration equal to the systolicejection interval (as determined above) may be utilized in lieu of thetransient impulses.

Second, the relationship between the cardiac contractions signal (x(t))and the PAP waveform segment (y(t)) is determined by estimating both aconstant term and an impulse response (h(t)) which when convolved withx(t) “best” fits y(t) minus the constant term. The estimated constantterm represents the average LAP, while the estimated h(t) characterizesthe dynamical properties of the pulmonary arterial tree and is definedto represent the (scaled) PAP-LAP response to a single cardiaccontraction. The impulse response h(t) and average LAP may bespecifically estimated according to the following ARX equation withconstant term c:

$\begin{matrix}{{{y(t)} = {c + {\sum\limits_{k = l}^{m}{a_{k}{y\left( {t - k} \right)}}} + {\sum\limits_{k = l}^{n}{b_{k}{x\left( {t - k} \right)}}} + {e(t)}}},} & (4)\end{matrix}$

where e(t) is the unmeasured residual error, the parameters {a_(k),b_(k)} completely specify h(t), and m and n limit the number of theseparameters (model order) [43]. For a fixed model order, the parametersincluding c are estimated from x(t) and y(t) through the least-squaresminimization of the residual error, which has a closed-form solution[43]. The model order is determined by minimizing any criterion (e.g.,Minimum Description Length criterion) or method known in the art thatpenalizes for unnecessary parameters [43]. Prior to this estimationprocedure, x(t) and y(t) may be lowpass filtered in order to amplify thecontribution of long time scale energy such that the least squaresestimation is prioritized at these time scales. With the estimatedparameters {â_(k),{circumflex over (b)}_(k)} and ĉ, average LAP and h(t)are computed as follows:

$\begin{matrix}{{LAP} = {\hat{c}/\left( {1 - {\sum\limits_{k = l}^{m}{\hat{a}}_{k}}} \right)}} & (5) \\{{h(t)} = {{\sum\limits_{k = l}^{n}{{\hat{a}}_{k}{h\left( {t - k} \right)}}} + {\sum\limits_{k = l}^{n}{{\hat{b}}_{k}{{\delta \left( {t - k} \right)}.}}}}} & (6)\end{matrix}$

Note that any other parametric model (e.g., autoregressive movingaverage with exogenous input (ARMAX) equation [43]) may be employed invarious embodiments of the invention to represent the structure of h(t),and any other minimization criterion (e.g., absolute error) may beutilized to determine the “best” h(t) and average LAP.

Third, the Windkessel time constant s of the pulmonary arterial tree isdetermined over the interval of h(t) for which the faster wave andinertial effects have dissipated For example, the interval ranging fromone to two seconds following the time of the maximum value of h(t) maybe selected. (See FIG. 5 and discussion above for other exemplary timeintervals.) This step may be achieved based on Equation (3) as describedabove. Thus, in certain embodiments of the invention, accuratedetermination of τ as well as LAP are achieved by virtue of h(t)coupling the long time scale variations in x(t) to y(t)-LAP.

Finally, proportional CO is computed via Ohm's law, and proportional PVRis given as T. Alternatively, when the area of each impulse in x(t) isset to an arbitrary constant value, proportional CO may be determined asthe product of the peak value of the estimated h(t) and the averageheart rate and proportional PVR may be subsequently determined via Ohm'slaw. The proportional, continuous CO and PVR estimates may becalibrated, if desired, as described above.

The aforementioned embodiments of the mathematical analysis may also beapplied to a SAP waveform (or a related waveform such as aphotoplethysmography signal) measured at any site in the systemicarterial tree. In this way, proportional CO, proportional SVR, andaverage SVP or only proportional CO and proportional SVR (with SVPassumed to be negligible, i.e., c=0) may be continuously monitored.

In another embodiment, proportional CO, average LAP, and otherhemodynamic variables are determined by mathematical analysis of thecontour of a RVP waveform. A key idea of this aspect of the invention isthat RVP and PAP may be regarded as nearly equal during the systolicejection interval of each beat provided that there is no significantstenosis of the pulmonic valve (FIG. 6). The systolic ejection intervalsof the observed RVP waveform are therefore identified so as to producean “incomplete” PAP waveform without diastolic pressure information.Then, with this incomplete PAP waveform, the clinically significanthemodynamic variables may be determined similarly to embodimentsdescribed above.

For example, FIG. 6 illustrates one such embodiment of the mathematicalanalysis, which is applied to a segment of a digitized RVP waveform(sampled at, e.g., 90 Hz) of duration greater than a cardiac cycle(e.g., 30 seconds to ten minutes). This particular embodiment isemployed in six steps.

First, each systolic ejection interval within the RVP waveform segmentis identified. The beginning of systole for each beat is determined asthe time of the maximum temporal derivative of each RVP pulse [61],while the end of systole for each beat is identified as the time of thepeak value of each RVP pulse. Alternatively, any other method known inthe art (e.g., see above) may be used to identify the systolic ejectionintervals. This step produces an incomplete segment of a PAP waveform.

Second, a cardiac contractions signal is constructed as described above.Note that the resulting cardiac contractions signal is defined for allvalues of time (i.e., both systole and diastole).

Third, the relationship between the complete cardiac contractions signal(x(t)) and the incomplete PAP waveform (y(t)) is characterized byestimating both a constant term representing average LAP and the impulseresponse (h(t)) which when convolved with x(t) “best” fits y(t) minusthe average LAP. The impulse response h(t) and average LAP may bespecifically estimated according to the following output error equationwith constant term c:

$\begin{matrix}{{{w(t)} = {{\sum\limits_{k = l}^{m}{a_{k}{w\left( {t - k} \right)}}} + {\sum\limits_{k = l}^{n}{b_{k}{x\left( {t - k} \right)}}}}}{{y(t)} = {{w(t)} + c + {{e(t)}.}}}} & (7)\end{matrix}$

Again, e(t) is the unmeasured residual error, the parameters {a_(k),b_(k)} completely specify h(t), and m and n limit the number of theseparameters (model order) [43]. For a fixed model order, the parametersincluding c are estimated from x(t) and the incomplete y(t) through theleast-squares minimization of the residual error during the systolicejections intervals. This optimization problem may be solved, forexample, through a numerical search procedure (e.g., Gauss Newtonmethod) or the Stieglitz-McBride iteration [43]. The model order may bedetermined similarly to that of Equation (1). Prior to this estimationprocedure, x(t) and y(t) may be lowpass filtered in order to amplify thecontribution of long time scale energy such that the least squaresestimation is prioritized at these time scales. With the estimatedparameters {â_(k),{circumflex over (b)}_(k)} and ĉ, h(t) is computedaccording to Equation (6) and average LAP is given as ĉ. Note that otherparametric models may be employed in various embodiments of theinvention to represent the structure of h(t), and any other minimizationcriterion (e.g., absolute error) may be utilized to determine the “best”h(t) and average LAP. However, some methods known in the art may be lesspreferable (e.g., linear least squares identification of ARX models),since the output is not known for all time here.

Fourth, τ is determined from the estimated h(t) based on Equation (3) asdescribed above. Again, in certain embodiments of the invention,accurate determination of τ as well as average LAP is achieved by virtueof h(t) coupling the long time scale variations in x(t) to theincomplete y(t)-LAP.

Fifth, the complete PAP waveform including diastolic information isconstructed by convolving the cardiac contractions signal x(t) with theestimated h(t) and then summing this response (z(t)) to the average LAP.This step is mathematically achieved as follows:

$\begin{matrix}{{{w(t)} = {{\sum\limits_{k = l}^{m}{{\hat{a}}_{k}{w\left( {t - k} \right)}}} + {\sum\limits_{k = l}^{n}{{\hat{b}}_{k}{x\left( {t - k} \right)}}}}}{{y(t)} = {{w(t)} + {\hat{c}.}}}} & (8)\end{matrix}$

Finally, proportional CO is determined via Ohm's law, and proportionalPVR is trivially obtained with τ. Alternatively, when the area of eachimpulse in x(t) is set to an arbitrary constant value, proportional COmay also be determined as the product of the peak value of the estimatedh(t) and the average heart rate and proportional PVR may be subsequentlydetermined via Ohm's law. Again, if desired, the proportional,continuous CO and PVR estimates may be calibrated as described above.This embodiment of the invention may also be applied to LVP waveforms inorder to continuously monitor proportional CO, proportional SVR, andaverage SVP or only proportional CO and proportional SVR (with SVPassumed to be negligible, i.e., c=0). The invention includes apparatusfor performing the methods described above, e.g., for determining LAP.For example, the apparatus may comprise memory means that stores aprogram comprising computer-executable process steps and a processingunit that executes the computer-readable process steps so as to performa mathematical analysis of the contour of a PAP waveform, RVP waveform,SAP waveform, LVP waveform or another circulatory signal (or any two ormore of the foregoing) and uses the mathematical analysis to determineaverage LAP, proportional CO, proportional PVR, proportional SVR,average SVP, etc. The apparatus may perform any one or more of themethods described herein to determine any one or more of the hemodynamicvariables discussed herein, e.g., average LAP, proportional CO,proportional PVR, proportional SVR, average SVP.

In certain embodiments of the present invention, an analog PAP waveform,RVP waveform, SAP waveform, LVP waveform or another circulatory signalis fed into an analog-to-digital converter as it is being measured. Thecirculatory signal may be acquired using standard methods, such as thosementioned above. In certain embodiments of the invention, a surface ECGis also measured, e.g., via standard ECG leads. The digitizedcirculatory signal and the ECG, if measured, are stored in a buffersystem. The most recent time intervals of the digitized signals (e.g.,one cardiac cycle to ten minutes) are transferred from the buffer systemto a processing unit, which analyzes the signal according to the methodsof the invention. The buffer and processing unit may be implementedusing, for example, any standard microcomputer or implanted circulatorymonitoring device running appropriate software to implement themathematical operations described above. The software components of theinvention may be coded in any suitable programming language and may beembodied in any of a range of computer-readable media including, but notlimited to, floppy disks, hard disks, CDs, zip disks, DVD disks, etc. Itwill be appreciated that the term “processing unit” is used herein torefer to any suitable combination of general and/or special purposeprocessors. For example, at least some of the processing steps may beimplemented using hardware, e.g., signal processing chips. Thus themethods may be implemented using any suitable combination of generaland/or special purpose processors and appropriate software.Computer-readable media that store computer-executable process steps toperform all or part of one or more methods of the invention are also anaspect of the invention.

Outputs such as proportional CO, LAP or SVP, and proportional PVR or SVRmay be displayed on a visual display such as a computer screen and/ormay be printed or transmitted to a remote location. The ECG, andanalysis thereof, may also be displayed. In a preferred embodiment ofthe system, the process is continuously repeated thereby providing theon-line monitoring of proportional CO, LAP or SVP, proportional PVR orSVR, and/or other hemodynamic variables (e.g., with a delay equal tohalf the selected analysis interval). Alternatively or additionally,absolute CO and PVR or SVR may be computed and displayed through anomogram or a single, independent measurement of absolute CO. In certainembodiments of the invention, an alarm is triggered upon excessivechanges in any of the estimated variables.

Finally, the methods may further comprise the step of selecting,recommending, suggesting, or administering a therapy to the subject, ormodifying the subject's therapy, based on values for one or morehemodynamic variables obtained according to the methods and apparatus.The therapy may be, for example, any medical or surgical therapy knownin the art that is suitable for treating a subject having aphysiological state consistent with the determined average LAP, averageSAP, proportional CO, proportional PVR and/or proportional SVR.Exemplary therapies include pharmaceutical agents, e.g., pressors ordiuretics, fluids, etc. In certain embodiments of the invention therapyis administered or ongoing therapy is modified (e.g, the dose of apharmaceutical agent is increased or decreased) automatically. Forexample, the apparatus may interface with or include a device thatmodifies the infusion rate of a solution (optionally containing atherapeutic agent) in response to values for one or more of thehemodynamic variables measured according to the invention. Byintermittently or continuously determining values for one or morehemodynamic variables and adjusting the infusion rate and/or dose of atherapeutic agent accordingly, the inventive system can operate usingfeedback, e.g., to provide closed-loop therapy. Specific desired values,or ranges of desired values, for the hemodynamic variables can beselected, e.g., by a patient's health care provider, and the apparatuscan modify therapy in order to achieve these values. In certainembodiments of the invention the apparatus monitors and optionallyrecords, stores, and/or analyzes the response of one or more of thehemodynamic variables to changes in therapy. The analysis can be used tomake further adjustments or modifications to the therapeutic regimen.For example, the system can learn how particular therapies oralterations in therapy affect one or more hemodynamic variables and canselect future therapy accordingly. In certain embodiments of theinvention, suggested or recommended therapies or modifications toongoing therapies are displayed on a display device.

A system or apparatus of the present invention may also include meansfor acquiring a circulatory signal from a subject. For example, a systemor apparatus of the invention may include a catheter, e.g., a pulmonaryartery catheter. If desired, certain of the signal processing steps maybe performed by the acquisition means itself. For example, theacquisition means may include hardware for performing filtering,analog-to-digital conversion, analysis, etc., of the acquired signal.

The foregoing description is to be understood as being representativeonly and is not intended to be limiting. Alternative systems andtechniques for implementing the methods of the invention will beapparent to one of skill in the art and are intended to be includedwithin the accompanying claims. It will be appreciated that wherever theclaims recite a method for determining one or more hemodynamicvariables, the invention comprises apparatus and computer-executableprocess steps for performing the steps of the method and also comprisesa method for monitoring a subject using the method for determining oneor more hemodynamic variables and further comprises a method of treatinga subject comprising monitoring a subject using the method andrecommending, suggesting, selecting, modifying, and/or administering atherapy based at least in part on data and/or information acquired orobtained using the method, e.g., based at least in part on the subject'saverage LAP, average SVP, proportional CO, proportional PVR, and/orproportional SVR as determined according to the method. In addition, themethods and/or apparatus of the invention may be used to determine ormonitor any one or more of these hemodynamic variables, e.g., any subsetof these hemodynamic variables. All references cited herein areincorporated by reference. In the event of a conflict or inconsistencybetween any of the incorporated references and the instantspecification, the specification shall control, it being understood thatthe determination of whether a conflict or inconsistency exists iswithin the discretion of the inventors and can be made at any time.

Those skilled in the art will recognize, or be able to ascertain usingno more than routine experimentation, many equivalents to the specificembodiments of the invention described herein. The scope of the presentinvention is not intended to be limited to the above Description, butrather is as set forth in the appended claims. In the claims, articlessuch as “a,”, “an” and “the” may mean one or more than one unlessindicated to the contrary or otherwise evident from the context. Claimsor descriptions that include “or” between one or more members of a groupare considered satisfied if one, more than one, or all of the groupmembers are present in, employed in, or otherwise relevant to a givenmethod, apparatus, etc. unless indicated to the contrary or otherwiseevident from the context. Furthermore, it is to be understood that theinvention encompasses all variations, combinations, and permutations inwhich one or more limitations, elements, clauses, descriptive terms,etc., from one or more of the listed claims is introduced into anotherclaim. In particular, any claim that is dependent on another claim canbe modified to include one or more limitations found in any other claimthat is dependent on the same base claim. In addition, it is to beunderstood that any one or more specific embodiments of the presentinvention may be explicitly excluded from the claims.

REFERENCES

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1. A method for determining the asymptotic value of the PAP waveform were cardiac contractions to cease comprising: performing a mathematical analysis of the PAP waveform to determine the asymptotic value.
 2. The method of claim 1, further comprising the step of: selecting the asymptotic value as the value of LAP.
 3. The method of claim 1, wherein the mathematical analysis comprises analyzing temporal variations of at least an interval of the PAP waveform within a cardiac cycle.
 4. The method of claim 1, wherein the mathematical analysis comprises analyzing temporal variations of at least an interval of the PAP waveform from one cardiac cycle to another.
 5. The method of claim 1, wherein the mathematical analysis comprises analyzing temporal variations of at least an interval of the PAP waveform both within a cardiac cycle and from one cardiac cycle to another.
 6. The method of claim 1, wherein the method comprises steps of: (a) fitting the sum of one or more basis functions, at least one of which is or includes a constant term, to one or more individual diastolic decay intervals of the PAP waveform over one or more cardiac cycles; and (b) selecting the constant term as the asymptotic value.
 7. The method of claim 6, further comprising the step of: (c) selecting the asymptotic value as the value of LAP.
 8. The method of claim 6, wherein at least some of the basis functions are complex exponential functions.
 9. The method of claim 1, further comprising determining proportional CO, proportional PVR, or both.
 10. The method of claim 9, wherein proportional CO is computed by: (a) determining the Windkessel time constant of the pulmonary arterial tree; and (b) computing the mean value of PAP minus LAP over one or more cardiac cycles, wherein LAP is given by the asymptotic value of PAP; and (c) dividing the result of step (b) by the time constant, thereby determining proportional CO.
 11. The method of claim 10, wherein the time constant is computed by steps of: (a) fitting the sum of one or more basis functions, at least one of which is or includes a constant term, to the diastolic decay interval of the PAP waveform over multiple cardiac cycles; (b) extrapolating the computed basis functions to low-pressure values; and (c) fitting a mono-exponential function to the extrapolated pressure values at times such that the faster wave and inertial effects have dissipated, wherein the mono-exponential function provides the Windkessel time constant of the pulmonary arterial tree.
 12. The method of claim 1 1, wherein at least some of the basis functions are complex exponential functions.
 13. The method of claim 9, wherein proportional PVR is computed by: (c) determining the Windkessel time constant of the pulmonary arterial tree; and (d) selecting the time constant as the value for proportional PVR.
 14. The method of claim 9, further comprising the step of: obtaining absolute values for CO or PVR by calibrating the computed proportional CO or PVR to an absolute CO value or PVR value, respectively, obtained from an absolute CO measurement.
 15. The method of claim 1, wherein the method comprises steps of: (a) fitting the sum of (i) a constant term and (ii) the convolution of an impulse response characterizing the pulmonary arterial tree with a cardiac contractions signal, to a segment of a PAP waveform of duration greater than a cardiac cycle; and (b) selecting the constant term as the asymptotic value.
 16. The method of claim 15, further comprising selecting the asymptotic value as the value of LAP.
 17. The method of claim 15, wherein the impulse response represents the response of PAP minus LAP to a single cardiac contraction.
 18. The method of claim 15, wherein the cardiac contractions signal comprises an impulse train or a train of pulses of any shape.
 19. The method of claim 15, further comprising the step of determining proportional CO by: (c) determining the Windkessel time constant of the pulmonary arterial tree; and (d) computing the mean value of PAP minus LAP over one or more cardiac cycles; and (e) dividing the result of step (d) by the time constant, thereby determining proportional CO.
 20. The method of claim 19, wherein the time constant is determined by the step of: fitting an exponential function to the tail end of the impulse response at times such that the faster wave and inertial effects have dissipated, wherein the exponential function provides the Windkessel time constant of the pulmonary arterial tree.
 21. The method of claim 15, further comprising the step of: determining proportional CO by multiplying the peak value of the impulse response of the pulmonary arterial tree by the average heart rate.
 22. A method for determining LAP comprising: performing a mathematical analysis of the PAP waveform to determine LAP.
 23. A method of determining one or more of average LAP, proportional CO, or proportional PVR, comprising: (a) performing a mathematical analysis of a PAP waveform to determine the asymptotic value of the PAP waveform were cardiac contractions to cease; and (b) using the asymptotic value to determine one or more of average LAP, proportional CO, or proportional PVR.
 24. The method of claim 23, further comprising the step of: obtaining absolute values for CO or PVR by calibrating the computed proportional CO or PVR to an absolute CO value or PVR value, respectively, obtained from an absolute CO measurement.
 25. A method for determining the asymptotic value of the RVP waveform were cardiac contractions to cease comprising: performing a mathematical analysis of the RVP waveform to determine the asymptotic value.
 26. The method of claim 25, wherein the method comprises the steps of: (a) constructing a partial PAP waveform from the RVP waveform; and (b) performing a mathematical analysis of the partial PAP waveform.
 27. The method of claim 25, wherein the method comprises the steps of: (a) obtaining the partial PAP waveform from systolic ejection intervals of the RVP waveform; and (b) performing a mathematical analysis of the partial PAP waveform.
 28. The method of claim 27, wherein the PAP waveform that includes systolic ejection intervals is obtained by: (i) identifying systolic ejection intervals of the RVP waveform; and (ii) utilizing the systolic ejection intervals as an approximation of the PAP waveform during the systolic ejection intervals.
 29. The method of claim 28, wherein the systolic ejection interval for each beat is determined by computing the interval between the time of the maximum temporal derivative of each RVP pulse and the time of the peak value of each RVP pulse.
 30. The method of claim 28, wherein the systolic ejection interval for each beat is determined by analyzing a surface ECG, a phonocardiogram, or both.
 31. The method of claim 26, wherein the method comprises steps of: (c) fitting the sum of (i) a constant term and (ii) the convolution of an impulse response characterizing the pulmonary arterial tree with a cardiac contractions signal, to a partial PAP waveform of duration greater than a cardiac cycle; and (d) selecting the constant term as the asymptotic value of the PAP waveform.
 32. The method of claim 31, further comprising selecting the asymptotic value as the value for LAP.
 33. The method of claim 31, wherein the impulse response represents the response of PAP minus LAP to a single cardiac contraction.
 34. The method of claim 31, wherein the cardiac contractions signal comprises an impulse train or train of pulses of any shape.
 35. The method of claim 31, comprising constructing a PAP waveform including both systolic and diastolic intervals by convolving the cardiac contractions signal with the impulse response and adding the constant term.
 36. The method of claim 31, wherein the impulse response of the pulmonary arterial tree represents the response of PAP minus LAP to a single cardiac contraction, the method further comprising the step of determining proportional CO by: (e) determining the Windkessel time constant of the pulmonary arterial tree; (f) computing the mean value of the complete PAP waveform minus LAP over multiple cardiac cycles, wherein LAP is given by the asymptotic value of RVP; and (g) dividing the result of step (f) by the time constant, thereby determining proportional CO.
 37. The method of claim 31, wherein the impulse response of the pulmonary arterial tree represents the response of PAP minus LAP to a single cardiac contraction, the method further comprising the step of determining proportional CO by: (e) determining the Windkessel time constant of the pulmonary arterial tree; (f) constructing a PAP waveform that includes both systolic and diastolic intervals from intervals of the PAP waveform; (g) computing the mean value of PAP minus LAP over one or more cardiac cycles; and (h) dividing the result of step (g) by the time constant, thereby determining proportional CO.
 38. The method of claim 37, wherein the PAP waveform that includes both systolic and diastolic intervals is constructed by adding the constant term to the convolution of the cardiac contractions signal with the impulse response.
 39. The method of claim 37, wherein the time constant is computed by the step of: fitting an exponential function to the tail end of the impulse response at times such that the faster wave and inertial effects have dissipated, wherein the exponential function provides the Windkessel time constant of the pulmonary arterial tree.
 40. The method of claim 31, further comprising the step of: determining proportional CO by multiplying the peak value of the impulse response of the pulmonary arterial tree and the average heart rate.
 41. The method of claim 25, wherein proportional PVR is computed by: (a) determining the Windkessel time constant of the pulmonary arterial tree; and (b) selecting the time constant as the value for proportional PVR.
 42. The method of claim 25, further comprising determining proportional CO, proportional PVR, or both.
 43. The method of claim 25, further comprising the step of: obtaining absolute values for CO or PVR by calibrating the computed proportional CO or PVR to an absolute CO value or PVR value, respectively, obtained from an absolute CO measurement.
 44. A method for determining LAP comprising: performing a mathematical analysis of the RVP waveform to determine LAP.
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 95. An apparatus for determining the asymptotic value of the PAP waveform, the asymptotic value of the RVP waveform, the asymptotic value of the SAP waveform, or the asymptotic value of the LVP waveform, were cardiac contractions to cease comprising: (a) memory means that stores a program comprising computer-executable process steps; and (b) a processing unit that executes the computer-readable process steps so as to perform a mathematical analysis of any one or more of a PAP, RVP, SAP, or LVP waveform and determines the asymptotic value of the PAP waveform, the asymptotic value of the RVP waveform, the asymptotic value of the SAP waveform, or the asymptotic value of the LVP waveform, were cardiac contractions to cease.
 96. The apparatus of claim 94, comprising input means to accept a signal representing a PAP, RVP, LVP, or SAP waveform.
 97. The apparatus of claim 94, comprising means for acquiring a PAP, RVP, SAP, or LVP waveform from a subject.
 98. The apparatus of claim 94, further comprising an analog-to-digital converter.
 99. The apparatus of claim 94, wherein the apparatus comprises a buffer system.
 100. The apparatus of claim 94, wherein the apparatus comprises a display device.
 101. The apparatus of claim 94, wherein the apparatus executes one or more steps to select the asymptotic value of PAP or RVP as the value for average LAP or selecting the asymptotic value of SAP or LVP as the value for average SVP.
 102. The apparatus of claim 94, wherein the apparatus executes one or more steps to determine proportional CO, proportional PVR, and/or proportional SVR using the asymptotic value of the PAP or RVP waveform.
 103. A method of constructing a complete PAP waveform including both systolic and diastolic intervals from an RVP waveform comprising steps of: (a) obtaining a partial PAP waveform from the RVP waveform; and (b) performing a mathematical analysis of the partial PAP waveform to construct a PAP waveform including both systolic and diastolic intervals.
 104. The method of claim 103, wherein the partial PAP waveform is obtained by: (i) identifying systolic ejection intervals of the RVP waveform; and (ii) utilizing the systolic ejection intervals as an approximation of the PAP waveform during the systolic ejection intervals.
 105. The method of claim 104, wherein the systolic ejection interval for each beat is determined by computing the interval between the time of the maximum temporal derivative of each RVP pulse and the time of the peak value of each RVP pulse.
 106. The method of claim 104, wherein the systolic ejection interval for each beat is determined by analyzing a surface ECG, a phonocardiogram, or both.
 107. The method of claim 104, wherein the method comprises steps of: (c) fitting the sum of (i) a constant term and (ii) the convolution of an impulse response characterizing the pulmonary arterial tree with a cardiac contractions signal, to the partial PAP waveform; and (d) constructing a PAP waveform including both systolic and diastolic intervals by convolving the impulse response determined in step (c) with the cardiac contraction signal and adding the constant term.
 108. The method of claim 103, further comprising the step of computing the mean value of the PAP waveform including both systolic and diastolic intervals to obtain mean PAP.
 109. An apparatus for constructing a PAP waveform including both systolic and diastolic intervals from an RVP waveform comprising: (a) memory means that stores a program comprising computer-executable process steps; and (b) a processing unit that executes the computer-readable process steps so as to (i) obtaining a partial PAP waveform from the RVP waveform; and (ii) perform a mathematical analysis of the partial PAP waveform to construct a PAP waveform including both systolic and diastolic intervals.
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